A panoply of unconventional electronic states has been observed in moiré superlattices. Engineering similar bosonic phases remains, however, largely unexplored. We report the observation of a bosonic correlated insulator in tungsten diselenide/tungsten disulfide (WSe2/WS2) moiré superlattices composed of excitons, that is, tightly bound electron-hole pairs. We develop a pump probe spectroscopy method that we use to observe an exciton incompressible state at exciton filling νex = 1 and charge neutrality, indicating a correlated insulator of excitons. With varying charge density, the bosonic correlated insulator continuously transitions into an electron correlated insulator at charge filling νe = 1, suggesting a mixed correlated insulating state between the two limits. Our studies establish semiconducting moiré superlattices as an intriguing platform for engineering bosonic phases.